Rodrick Wallace1
(1)
New York State Psychiatric Institute, New York, NY, USA
Summary
Evolutionary process has selected for inherently unstable systems in higher animals that can react swiftly to changing patterns of threat or opportunity, for example, blood pressure, the immune response, and gene expression. However, these require continual strict regulation: uncontrolled blood pressure is fatal, immune cells can attack “self” tissues, and improper gene expression triggers developmental disorders. Consciousness in particular demands high rates of metabolic free energy to both operate and regulate the fundamental biological machinery: both the “stream of consciousness” and the “riverbanks” that confine it to socially defined useful realms are constructed and reconstructed moment-by-moment in response to highly dynamic internal and environmental circumstances. We develop powerful necessary conditions models for such phenomena based on the Data Rate Theorem linking control and information theories in the context of inherent instability. The synergism between conscious action and its regulation underlies the ten-fold higher rate of metabolic energy consumption in human neural tissues and implies a close, culturally modulated relation between sleep disorders and certain psychopathologies that may be induced by environmental stress, in a large sense.
5.1 Introduction
Why do neural tissues in humans consume metabolic free energy at ten times the rate of other tissues (Clarke and Sokoloff 1999)? A simplistic answer is that, in humans, consciousness must operate with a time constant of a hundred milliseconds, and straightforward adaptation of the Arrhenius reaction rate law suggests an exponential increase of neural reaction rate with the rate of adenosine triphosphate (ATP) consumption. While this surely accounts for substantial increase in energy demand as we will argue below, an order of magnitude difference in energy consumption seems somewhat excessive. Is more going on? We will contend that, in addition to the matter of a short time constant, metabolic free energy must be supplied to parallel, rapidly operating regulatory systems that stabilize consciousness as an inherently unstable phenomenon constrained by the Data Rate Theorem, necessitating independent metabolic free energy supplies for control purposes.
A control theory digression. At the beginning of World War II, according to report, British fighters were close adaptations of training aircraft that were inherently stable, in the sense that the aerodynamic center of pressure (CP) was well behind the machine’s center of gravity (CG). Thus, hands-off, a perturbed aircraft would, after a few oscillations, return to stable flight. Early German fighters had less separation between CG and CP and were far harder to fly, but, in consequence, could turn on a dime, and were significantly better in high speed combat maneuvers than early British fighters, with serious consequences for the Allies. Current fighter jets are, by contrast, designed to be inherently unstable, thus even more highly maneuverable, and must be flown-by-wire using a number of independent computers operating under high speed majority-voting rules. For such machines, regulation is everything.
Indeed, for many physiological systems, regulation is likewise—almost—everything. Some examples.
Even our basic multicellularity seems inherently unstable: cancerous “cheating” is expected to be an ongoing threat to multi-celled organisms (Aktipis et al. 2015). Nunney (1999) looked at cancer occurrence as a function of animal size, arguing that, in larger animals, whose lifespan grows as about the 4/10 power of their cell count, prevention of cancer in rapidly proliferating tissues becomes more difficult in proportion to size. Cancer control requires the development of additional mechanisms and systems to address tumorigenesis as body size increases—a synergistic effect of cell number and organism longevity.
This pattern may represent a real barrier to the evolution of large, long-lived animals, and Nunney predicts that those that do evolve have recruited additional controls over those of smaller animals to prevent cancer. Different tissues may have evolved markedly different tumor control strategies, all energetically expensive, using different complex signaling strategies, and subject to a multiplicity of reactions to signals, including, in social animals like humans, those related to psychosocial stress.
In a similar way, gene expression itself seems inherently unstable, requiring massive regulatory machineries at every stage of growth to prevent developmental disorders. Many such stages involve critical periods in which “epigenetic” factors—including patterns of cultural input and psychosocial stress often embedded in generational history—have amplified effect (Wallace and Wallace 2009, 2010).
The immune system, seen as an independent subcomponent of the more general tumor control system (Atlan and Cohen 1998), is also inherently unstable: failure of differentiation between “self” and “nonself” leads to carcinogenic chronic inflammation (Rakoff-Nahoum 2006) and autoimmune disorders (Mackay and Rose 2014). The immune system must, then, both respond quickly to injury or pathogenic challenge and yet be closely regulated to avoid self-attack.
On a shorter timescale, unregulated blood pressure would be quickly fatal in any animal with a circulatory system. The associated baroreceptor control reflex is not simple (Rau and Elbert 2001), but can be inhibited through peripheral processes, for example, under conditions of high metabolic demand. Higher brain structures modulate the reflex, for instance, when threat is detected and fight or flight responses are being prepared. This suggests, then, that blood pressure control is a broad and actively regulated modular physiological system.
The stream of consciousness, or whatever metaphor one prefers, seems similarly regulated, and high speed mechanisms, in concert with high speed regulators, will require a high rate of metabolic free energy.
Indeed, something of this has long been known.
Schiff (2008):
Many neurons within the central thalamus…share specific anatomical and physiological specializations that support their key role in the general functions of sustained attention, working memory, and motor preparation…Collectively, these anatomical specializations suggest that many neurons within the central thalamus may serve a general purpose function supporting large-scale cerebral dynamics associated with goal-directed behaviors and consciousness…Human imaging studies reveal that selective activation of central thalamus occurs both during tasks requiring short-term attention…and during tasks placing sustained demands of high vigilance over extended time periods…
Markman and Otto (2011):
…[C]haracterization of…optimality…should take into account the resource-limited nature of the human cognitive apparatus…the brain consumes a significant amount of energy. Thus energy minimization is likely to be an important constraint on cognitive processing…
The ideal observer or actor defined purely in terms of information is…a point of comparison against human cognitive or sensory abilities rather than as a statement of what is optimal as a cognitive process…A definition of optimal behavior needs to take energy minimization into account.
Christie and Schrater (2015):
[T]reating cognitive effort as a resource control problem…[implies] the brain has mechanisms that allow control of cognitive effort…[T]here are limitations on…the ability to control resource allocation…[and]…on the impact that effort has on performance…[I]t is reasonable to posit the existence of some signal by which neuronal gain is modulated according to both the availability of metabolic resource and reward signals.
Here we will attempt to formally address the deep synergistic intertwinings of metabolic energy rate, information transmission rate, and system control using powerful necessary conditions asymptotic limit theorems allowing construction of statistical models that can be fitted to data.
We will, in particular, examine the environmental induction of sleep pathologies in their inevitable cultural context.
5.2 Reaction Rate
Physiological processes such as wound healing, the immune response, tumor control, and animal consciousness all represent the evolutionary exaptation of inevitable information crosstalk into dynamic processes that recruit sets of simpler cognitive modules to build temporary working coalitions addressing particular patterns of threat and opportunity confronting an organism, as described in Chap. 1 Such tunable coalitions operate, however, at markedly different rates. Wound healing, depending on the extent of injury, may take 18 months to complete its work (Mindwood et al. 2004). Animal consciousness typically operates with a time constant of a few hundred milliseconds.
How can phenomena acting on such different rates be subsumed under the same underlying mechanism? A heuristic adaptation of Arrhenius’ law (Laidler 1987), which predicts exponential differences in reaction rate with “temperature,” in a large sense, produces a first approximation to the result, recognizing that, in contrast to simple chemical reactions, cognitive phenomena are inherently nonequilibrium. That is, a large class of cognitive processes can be associated with dual information sources for which palindromes are highly improbable. The easiest way to understand the association is to recognize that, at its most basic, cognition at any scale or level of organization requires choice of some particular action from a larger set of those possible. Such choice reduces uncertainty, and the reduction of uncertainty implies existence of an information source. The argument can be made quite formally, as in the first chapter.
The rate of biocognition, we nonetheless argue, appears driven by the rate of available metabolic free energy as something of a heuristic temperature analog. Absent other theory, we try the Arrhenius relation. A more different treatment, leading to similar results, can be found in Eqs. (11)–(15) and Fig. 4 of Kostal and Kobayashi (2015).
In any event, the energetics of biological reactions are remarkable: at 300 K, molecular energies represent approximately 2.5 kJ/mol in available free energy. By contrast, the basic biological energy reaction, the hydrolysis of adenosine triphosphate (ATP) to adenosine diphosphate—under proper conditions at 300 K, produces some 50 kJ/mol in reaction energy. This is equivalent to a “reaction temperature” of 6000 K. Increasing the rate of ATP delivery to one kind of tissue an order of magnitude over any others provides sufficient energy for very rapid biocognition.
The question is how such rapid biocognition is parceled out between consciousness itself and the mechanisms that must regulate and stabilize it.
In more detail, given a chemical reaction of the form aA + bB → pP + qQ, the rate of change in (for example) the concentration of chemical species P (written [P]) is often determined by an equation like
![$$\displaystyle{ d[P]/dt =\gamma (T)[A]^{n}[B]^{m} }$$](computational-psychiatry.files/image201.png){kind=small}
(5.1)
where n and m depend on the reaction details. The rate constant γ is expressed by the Arrhenius relation as
![$$\displaystyle{ \gamma =\alpha \exp [-E_{a}/CT] }$$](computational-psychiatry.files/image202.png){kind=small}
(5.2)
where α is another characteristic constant, Ea is the reaction activation energy, T is the Kelvin temperature, and C a universal constant. exp[−Ea ∕CT] is, using the Boltzmann distribution, the fraction of molecular interactions having energy greater than Ea .
The inherently nonequilibrium nature of cognition, however, requires a slightly more sophisticated treatment.
Consciousness appears to be largely an all-or-nothing phenomenon (Sergeant and Dehaene 2004), so that conscious signal perception must exceed a threshold before becoming entrained into the characteristic general broadcast.
A direct information theory argument focuses on the Rate Distortion Function (RDF) R(D) associated with the channel connecting the cognitive individual with an embedding and embodying environment. R(D) ≥ 0, a convex function (Cover and Thomas 2006), defines the minimum rate of information transmission needed to ensure that the average distortion between what is sent and what is received is less than or equal to D ≥ 0, according to an appropriate distortion measure. See Cover and Thomas (2006) for details. Assuming a threshold R0 for conscious perception of an incoming signal, we can, following Feynman’s (2000) identification of information as a form of free energy, write a Boltzmann-like probability for the rate of cognition as
![$$\displaystyle{ P[R \geq R_{0}] = \frac{\int _{R_{0}}^{\infty }\exp [-R/\omega M]dR} {\int _{0}^{\infty }\exp [-R/\omega M]dR} =\exp [-R_{0}/\omega M] }$$](computational-psychiatry.files/image203.png)
(5.3)
where M is the supplied rate of metabolic free energy, ω a constant (representing entropic loss-in-translation), and Fig. 5.1 follows, writing R0∕ω = k.
Fig. 5.1
The dashed lines are the “Arrhenius” relation exp[−k∕M] for rate of cognition as a function of the rate of available metabolic free energy M. The solid lines show the efficiency measure of rate per unit metabolic energy, exp[−k∕M]∕M. An order of magnitude increase in such free energy can enable several orders of magnitude increase in the rate of cognition. The point of greatest efficiency is at M = k, which we shall later interpret as an NREM sleep state in higher animals. The width of the efficiency measure, however, rapidly increases with k. Decline in M much below the shoulder of the curve triggers catastrophic collapse of cognition. The efficiency curve has the same overall form as the analogous Fig. 4 of Kostal and Kobayashi (2015), who use numerical methods
If we define an efficiency measure as (cognition rate)/M, we see that energy efficiency peaks at M = k, with the width of the curve depending on k; larger values generate very broad efficiency curves. The form of exp[−k∕M]∕M is distinct, and closely similar to what has been found in other recent work. Using numerical models of optimal coding and information transmission in Hodgkin–Huxley neurons under metabolic constraints, Kostal and Kobayashi (2015) find an almost exactly similar efficiency curve. Their treatment, however, involves regimes determined by the critical value of the effective reversal potential of an explicit neural model.
We will later interpret the efficiency peaks in Fig. 5.1 as characterizing low blood flow NREM sleep states in some higher animals.
For mammals, since body temperature remains constant, the rate of available metabolic free energy—dependent on mitochondrial function—serves as a temperature index for rates of biocognition. This determines the characteristic rate of chemically generated consciousness, or of the individual lower level cognitive modules that come together in a temporary assemblage to form such an analog. Neural tissues, in humans consuming metabolic energy at an order of magnitude greater rate than other tissues, thus can provide cognitive function many orders of magnitude faster than similar physiological phenomena.
But is this the whole story? Kostal and Kobayashi (2015) argue that efficiency matters in neural process, so that regimes of lower energy consumption may be favored over the highest cognitive rates. But, as Ristroph et al. (2013) argue, inherent instability, in itself, allows extremely rapid responses that have been strongly selected for. Here, we will argue that regulation of such phenomena must consume significant metabolic free energy, in addition to that needed for (relatively) rapid cognition.
How do we understand the regulation of inherently unstable control systems? Two fundamental relations, the Data Rate and Rate Distortion Theorems, are a necessary foundation. Their convolution, we shall show, provides further insight on metabolic energy demands of high speed cognition.
5.3 The Data Rate Theorem
The Data Rate Theorem (DRT), based on an extension of the Bode Integral Theorem for linear control systems, describes the stability of feedback control under data rate constraints (Nair et al. 2007). Given a noise-free data link between a discrete linear plant and its controller, unstable modes can be stabilized only if the feedback data rate 
is greater than the rate of “topological information” generated by the unstable system.
For the simplest incarnation—a linear expansion near a nonequilibrium steady state—the matrix equation of the plant is
(5.4)
where xt is the n-dimensional state vector at time t, ut is the imposed n-dimensional control signal vector at time t, Wt is an added noise signal, and A and B are, in this approximation, n × n fixed matrices. See Fig. 5.2 for a schematic. Then the necessary condition for stabilizability is
![$$\displaystyle{ \mathcal{H}>\log [\vert det(\mathbf{A}^{u})\vert ] \equiv a_{ 0} }$$](computational-psychiatry.files/image206.png){kind=small}
(5.5)
where det is the determinant and Au is the decoupled unstable component of A, i.e., the part having eigenvalues ≥ 1. The determinant represents a generalized volume. Thus there is a critical positive data rate below which there does not exist any quantization and control scheme able to stabilize an unstable system (Nair et al. 2007).
Fig. 5.2
“Regression model” for a control system near a nonequilibrium steady state. xt is system output at time t, ut the control signal, and Wt an added noise term
The DRT, in its various forms, relates control theory to information theory and is as fundamental as the Shannon Coding and Source Coding Theorems, and the Rate Distortion Theorem for understanding complex biological phenomena. In the next section we derive a more biologically relevant form of the DRT than Eq. (5.5) that relates control and stability to the availability of metabolic free energy.
Accepting Feynman’s (2000) insight that information is simply a form of free energy, in biological circumstances, we can write that 
, where M is the rate of metabolic free energy used to generate the control information rate 
, and m is a sharply increasing monotonic function, a consequence of massive entropic losses necessarily associated with translation of metabolic energy to information. Equation (5.5) thus implies that there is a minimum necessary rate of free energy consumption below which it is not possible to stabilize an inherently unstable biological control system. Some calculation provides details.
5.4 Rate Distortion Dynamics
We examine how the control signal ut in Fig. 5.2 is expressed in the system response xt+1. That is, we suppose it possible to deterministically retranslate—without error—a sequence of observed system outputs, Xi = x 1 i , x2 i …, at times 1, 2, …, into a sequence of control signals 
and then compare that inferred sequence with the original control signal sequence Ui = u 1 i , u2 i , …. The difference between them is a real number measured by some chosen measure of distortion, 
, in the sense of the Rate Distortion Theorem (Cover and Thomas 2006), allowing definition of an average distortion as
(5.6)
where p(Ui ) is the probability of the sequence Ui . We can then define a Rate Distortion Function (RDF), R(D), denoting the minimum channel capacity necessary for the limit condition < d > ≤ D to apply, often found using Lagrange multiplier methods or their generalizations (Cover and Thomas 2006). It is important to note that R(D) is always a convex function of D, i.e., a reverse J-shaped curve, allowing deep inference (e.g., Ellis 1985).
This allows invocation of a dynamic model as follows, adopting Feynman’s (2000) characterization of information as a form of free energy, measured here by the minimum channel capacity.
For a Gaussian channel, which has a Rate Distortion Function R(D) = 1∕2log[σ 2∕D], we can now define a “Rate Distortion entropy” as the Legendre transform of the RDF as
![$$\displaystyle{ S_{R} = R(D) - DdR(D)/dD = 1/2\log [\sigma ^{2}/D] + 1/2 }$$](computational-psychiatry.files/image213.png){kind=small}
(5.7)
The simplest nonequilibrium Onsager equation describing the dynamics of average distortion in terms of the gradient of this “entropy” is then (de Groot and Mazur 1984)
(5.8)
where t is the time and μ > 0 is a diffusion coefficient. This has the solution
(5.9)
which is the classic outcome of the diffusion equation. Such correspondence reduction serves as the foundation for arguing upward in both scale and complexity.
Other channels will have similar results as a consequence of the convexity of the Rate Distortion Function, i.e., monotonic growth in time. For example, a simple calculation shows that the “natural” channel, with R(D) ∝ 1∕D, gives 
.
Regulation, however, does not involve the diffusive drift of average distortion. Let M be the rate of metabolic free energy available for such regulation. Then a plausible model, in the presence of an internal system noise β2 in addition to the environmental channel noise defined by σ 2, is the stochastic differential equation
(5.10)
where dWt represents unstructured white noise and F(M) ≥ 0 is a monotonically increasing function in the rate of metabolic free energy supply M.
This relation has the nonequilibrium steady state expectation
(5.11)
Using the Ito chain rule on equation (5.10) (Protter 1990; Khasminskii 2012), it is possible to calculate the variance in the distortion as E(Dt 2) − (E(Dt ))2.
Applying the Ito relation Dt 2, however, we find that no real number solution for its expectation is possible unless the discriminant of the resulting quadratic equation is ≥ 0, giving the necessary condition
(5.12)
Note that similar conditions will apply to other kinds of channel as a consequence of the convexity of the RDF.
From Eq. (5.11), for a Gaussian channel,
![$$\displaystyle{ R_{nss} \geq \frac{1} {2}\log \left [\frac{\sigma ^{2}\beta ^{2}} {\sqrt{\mu }}\right ] }$$](computational-psychiatry.files/image220.png)
(5.13)
Applying the Black–Scholes calculation of the Mathematical Appendix to find the “cost” of R in terms of the rate M gives
![$$\displaystyle{ M = \frac{2C} {b^{2}} \log [R] +\kappa _{1}R +\kappa _{2} }$$](computational-psychiatry.files/image221.png)
(5.14)
where b is a noise term characteristic of the Black–Scholes approximation.
Substituting the result of equation (5.13) into this relation, a necessary condition for second order stability is
![$$\displaystyle\begin{array}{rcl} M_{nss}& \geq & \frac{2C} {b^{2}} \log \left [\frac{1} {2}\log \left [\frac{\sigma ^{2}\beta ^{2}} {\sqrt{\mu }}\right ]\right ] \\ & & +\kappa _{1}\frac{1} {2}\log \left [\frac{\sigma ^{2}\beta ^{2}} {\sqrt{\mu }}\right ] +\kappa _{2} \equiv a_{0}{}\end{array}$$](computational-psychiatry.files/image222.png)
(5.15)
where all constants are positive or zero.
This represents, for a Gaussian channel, another form of the Data Rate Theorem, involving an example of the iterated logarithm so characteristic of diffusion processes.
Values of M below this limit will trigger a phase transition into a disintegrated, pathological, system dynamic in a highly punctuated manner. Again, similar models can be constructed using the “natural” channel having the Rate Distortion Function R(D) = β∕D.
Note, however, the two distinctly different “cost” modes implied by the conditions κ1 = 0 , > 0.
Under more complex circumstances—for example, when there are several or many possible different F(M) functions representing different physiological and/or psychological states—setting the time-average expectation of dDtin Eq. (5.10) to zero
(5.16)
defines an index theorem in the sense of Hazewinkel (2002) and Atiyah and Singer (1963). An index theorem is an analytic relation whose solutions represent different topological modes of an underlying manifold, in a large sense. Such topologies are always characterized by group or groupoid structures (e.g., Lee 2000). The topologically distinct multiple solutions represent quasi-stable nonequilibrium steady state (nss) modes of the higher order maintenance system. These may range from a set of distinct nss fixed points to closed “Red Queen” cycles or pseudorandom “strange attractors” within a bounded region. Below, we will examine “directed transitions” between such modes representing large deviations in the sense of Champagnat et al. (2006), involving methods that can be applied when noise has “color.”
A somewhat different attack is possible, however, having implications for the regulation and control of sleep/wake dynamics.
5.5 The Sleep Cycle: An Optimization Model
Something much like the set of quasi-stable nss states implied by the index theorem of equation (5.16)—and by the κ1 = 0, > 0 conditions in equation (5.15)—can also be derived via an optimization argument applied to the rate calculation of equation (5.3). The essential point is that both consciousness and its necessary cognitive regulatory system(s) will follow similar metabolic scaling functions, so that we can seek to maximize a joint efficiency measure subject to constraint, applying the usual Lagrange multiplier argument. That is, letting the subscript C represent consciousness and R its regulatory machinery, we seek to maximize an efficiency functional
![$$\displaystyle{ \frac{\exp [-k_{C}/M_{C}]} {M_{C}} + \frac{\exp [-k_{R}/M_{R}]} {M_{R}} }$$](computational-psychiatry.files/image224.png)
(5.17)
subject to the constraint
(5.18)
The kX are appropriate constants and MX is the metabolic free energy rate for process X.
Taking
![$$\displaystyle\begin{array}{rcl} \boldsymbol{\Lambda }(M_{C},M_{R},\lambda )& \equiv & \frac{\exp [-k_{C}/M_{C}]} {M_{C}} + \frac{\exp [-k_{R}/M_{R}]} {M_{R}} \\ & & +\lambda (M_{C} + M_{R} - M) {}\end{array}$$](computational-psychiatry.files/image226.png)
(5.19)
gives the Lagrange optimization conditions as
(5.20)
The resulting complicated third order equation for solution pairs M C , MR implies the existence of several different possible optimization points for the system, strongly parameterized by the overall energy rate M. This represents another index theorem relating solutions of an analytic equation to underlying topological modes: increasing the number of systems to be optimized—adding terms of the form exp[−kJ ∕M J]∕MJ —increases the number of possible nonequilibrium quasi-stable states.
In a first approximation to a sleep cycle model, however, there are only a small number of stages to the normal pattern: NREM (non-rapid eye movement) which involves low rates of blood flow to the brain—low M—and REM sleep which can rival or exceed conscious state blood flows. Sleep states must, like other neural processes, be highly regulated, and the argument seems to carry through, as follows.
Assuming kC ≈ k R in Eq. (5.17), direct calculation shows a symmetric efficiency curve with one or two peaks, depending on the magnitude of M = MC + M R , as in Fig. 5.3. There, taking kX = 1, M C + MR = M = 1 → 3. 6, we obtain a maximum efficiency index, about 0.74, at MC = M R ≈ 1 for M ≈ 2, or along symmetric ridge points for larger M. The maximum value of 3.6 for M is chosen from the work of Madsen and Vorstrup (1991), who infer a 44% decline in cerebral metabolic rate during deep, slow-wave, NREM sleep.
Fig. 5.3
Total consciousness-and-regulator efficiency as a function of the metabolic free energy consumed by consciousness, MC , for kX = 1, M = M C + MR = 1 → 3. 6. A single maximum dominates the low energy mode, suggesting NREM sleep, denoted N, representing a 44% decline from the maximum M. The maxima at higher energies are ridges, suggesting REM sleep/awake modes, denoted R, W. Normal transitions are from W to N, followed by N-R oscillation. During sleep modes, “regulator” systems for consciousness may perform routine maintenance, as does the immune system when not fighting fires of infection, wound healing, or malignancy. NREM sleep becomes a transition path between the two more active states. Changing the values of the kX markedly shifts the relative heights and widths of the peaks, suggesting an added control mode. A little more work produces the models described in Booth and Diniz Behn (2014). There appears to be a necessary relation between sleep disturbance and the dysregulations of consciousness expressed as psychopathologies
That is, the highest efficiency peak is taken to be at the relatively quiescent NREM mode. For higher M, however, the two symmetric greatest efficiency points suggest waking/active sleep modes in which REM “sleep” may represent a parsimonious assumption of essential maintenance duties by systems otherwise dedicated to the regulation of awake consciousness. This seems analogous to the immune system which, when not extinguishing the fires of infection, wound healing, and malignancy, is deeply involved with processes of routine cellular maintenance (Cohen 2000). Long-term sleep dysfunction may be very serious indeed.
Similar ideas are in the literature. As Harris et al. (2012) put it,
Increasingly, sleep is thought to play an energetically restorative role in the brain…[D]uring sleep there is a transient increase in ATP level in cells of awake-active regions of the brain…suggested to fuel restorative biosynthetic processes in cells that, during the day, must use all their energy on electrical and chemical signaling. This implies an energy consumption trade-off; a high use of ATP on synapses during awake periods is balanced by more ATP being allocated to other tasks during sleep.
Energy use in the awake state also increases due to synaptic potentiation…compared to sleep…
These changes are reversed during sleep, presumably because of homeostatic plasticity…Thus sleep is essential for adjusting synaptic energy use.
It is known that the normal route from waking, the W point in Fig. 5.3, to REM sleep, the point R, is from W to N, and then down the other ridge from N to R. Direct transition from R to W does occur, and cycling between N and R is normal.
Note that values of kC = k R > 1 generate much broader symmetric curves with far less well-defined peaks, while values less than 1 are much more sharply peaked. Unequal values raise or lower one or the other ridge. Thus “tuning” these parameters would provide significant added system control.
This model is consistent with Hobson’s (2009, Fig. 4) three-dimensional Activation-Modulation-Input/Output (AIM) picture of the Western sleep cycle. In Hobson’s work, the normal progression is from waking to low-energy NREM sleep, followed by oscillation between NREM mode and the high-energy REM state.
Our model is also consistent with the observations of Hudson et al. (2014) on the punctuated recovery to consciousness from anesthesia. See their Fig. 4C which identifies three hubs between which there is directed transition during recovery.
REM sleep is now viewed as a state of consciousness which, compared to alert waking, is deficient in neuromuscular function (i.e., sleep paralysis), analytic ability, and transient episodic memory, while particularly rich in emotional cognition. In addition, both waking up and falling sleep involve complicated physiological processes to effect a transition between states. Since the late 1940s researchers have understood the importance of the reticular activating mechanism in the change from sleep to waking (Evans 2003). More recently, an arousal inhibitory mechanism, a thalamo-cortical process, has been recognized which transfers the body from waking to sleep (Evans 2003). Other work (Saper et al. 2005; Lu et al. 2006) has identified a putative neural flip-flop switch structure which controls REM sleep.
The standard phenomenological model for sleep is the “two-process” approach of Borbely (1982) and Daan et al. (1984). As Skeldon et al. (2014) put it, the homeostatic process takes the form of a relaxation oscillator that results in a monotonically increasing “sleep pressure” during wake that is dissipated during sleep. Switching from wake to sleep, and vice versa, occurs at upper and lower threshold values of the sleep pressure, respectively, where the thresholds are modulated by an approximately sinusoidal circadian oscillator. Skeldon et al. (2014) go on to show that the two-process model is essentially the same as the slow-time dynamic of the neurologically based Phillips and Robinson (2007) model.
But are there more general principles hidden here?
5.6 Transition Dynamics
How does a consciousness/regulator structure—however we choose to characterize it—make changes between the quasi-stable nss that the different modeling strategies above imply are central to the regulatory process? Indeed, recent primate experiments imply that even routine conscious decision making takes place in discrete steps (Latimer et al. 2015). Similar problems arise in evolutionary theory. Taking the approach of equation (5.10) in an evolutionary context, Champagnat et al. (2006) argue that the probability of a “large deviation” driving the system from one quasi-stable mode to another is given by a negative exponential of an entropy-like function
![$$\displaystyle{ \mathcal{I} = -\sum _{j}P_{j}\log [P_{j}] }$$](computational-psychiatry.files/image229.png)
(5.21)
where the Pj represent a particular probability distribution. This result—the large deviations argument—is well known in numerous contexts under various names—Sanov’s Theorem, the Gartner/Ellis Theorem, etc (Dembo and Zeitouni 1998). For the composite of human consciousness-and-regulation, we argue, the transition between “states” involves the effect of impinging information sources. That is, 
is not simply an “entropy” in this case, but represents action of an external information source (or sources) that, iteratively, regulates the internal regulators controlling individual consciousness.
A variant on this kind of approach would, for the optimization model, make different values of the essential parameters M, kC and k R the outputs of another, embedding, cognitive information source that drives the sleep cycle, turning off waking consciousness via the low-energy NREM mode, and then engaging high-energy REM sleep (Booth and Diniz Behn 2014).
5.7 Cultural Catalysis of the Sleep Cycle
As many have argued, humans sustain a dual heritage system of genes and culture (e.g., Richerson and Boyd 2006). Waking/REM consciousness and the associated full sleep cycle takes place in the context of a learned cultural system—an internalized information source having regularities of “grammar” and “syntax,” in a large sense—that must be maintained in memory, and such maintenance requires an additional expenditure of metabolic free energy beyond what is implied by Fig. 5.3. A simple heuristic argument follows from the “chain rule” of information theory (Cover and Thomas 2006) which states that, for information sources X and Y, the joint uncertainty H(X, Y ) follows an inequality:
(5.22)
We can, in general, write for the probability density function of some H at a metabolic free energy rate M,
![$$\displaystyle{ dP[H] = \frac{\exp [-H/kM]dH} {\int \exp [-H/kM]dH} }$$](computational-psychiatry.files/image232.png)
(5.23)
If k is very small as a consequence of an expected massive entropic loss, we can write, for the average of H,
![$$\displaystyle{ <H>= \frac{\int H\exp [-H/kM]dH} {\int \exp [-H/kM]dH} \approx kM }$$](computational-psychiatry.files/image233.png)
(5.24)
so that
(5.25)
If X represents the information source associated with the sleep/wake cycle, and Y that of internalized, learned culture, then we can expect that, through joint influence and crosstalk, culture will act as a kind of catalyst to canalize patterns of the sleep/wake cycle and their dysfunctions.
5.8 Environmental Induction of Sleep Disorders
We suppose, for the moment, that there is a scalar index of environmental stress, in a large sense, that affects patterns of sleep. Call it ρ. The stability condition of equation (5.15) then becomes something like
(5.26)
for appropriate functional forms in ρ. We make a first order approximation, i.e., M = κ1 ρ +κ2, f = κ3 ρ +κ4, so that the stability condition becomes
(5.27)
For small ρ, stability requires κ2∕κ 4 > a0. At high ρ the condition is κ1∕κ 3 > a0. If κ 2∕κ4 ≫ κ 1∕κ3 < a 0, then at some intermediate value of ρ, the basic inequality may be violated, leading to instability in the sleep/wake system. See Fig. 5.4.
Fig. 5.4
The horizontal line is the limit α0. If κ2∕κ 4 ≫ κ1∕κ 3 < a0, at some intermediate value of integrated environmental insult ρ, 
(κ3 ρ +κ4) falls below criticality, and control of the sleep/wake cycle fails
Rather than a simple scalar index of stress, there are usually multiple sources that interact via different mechanisms at various scales and levels of organization. Thus we are confronted by some matrix 
having components ρi, j ≠ ρj, i , i, j = 1, …, n since stresses may have different effects on each other. Square matrices of order n, however, have n scalar invariants, that is, n numbers that characterize the matrix regardless of how it may be expressed in different coordinate systems. The first is the trace, the last ± the determinant. The invariants are the coefficients of the characteristic polynomial of the matrix
(5.28)
where λ is a parameter, det is the determinant, and I is the n × n identity matrix.
We invoke the “Rate Distortion Manifold” methods of Glazebrook and Wallace (2009) or the “generalized retina” of Wallace and Wallace (2010) to argue that it is possible to accurately project high dimension information processes down onto low dimension “tangent spaces” that track an underlying complex information manifold, in a formal sense. Thus we assume there is some scalar function Γ of the matrix invariants ri that permits definition of a 
as in Eq. (5.27).
What are the dynamics of 
under stochastic circumstances? We again examine how a control signal ut in Fig. 5.2 is expressed in the system response xt+1. Again, we suppose it possible to deterministically retranslate a sequence of system outputs Xi = x1 i , x 2 i , … into a sequence of possible control signals 
and then compare that sequence with the original control sequence Ui = u 0 i , u1 i , …. The difference between them is a real number measured by a chosen distortion measure d, enabling definition of an average distortion < d > as in Eq. (5.6).
Again, we apply a classic Rate Distortion Theorem (RDT) argument. According to the RDT, there exists a Rate Distortion Function (RDF) that determines the minimum channel capacity, R(D), necessary to keep the average distortion < d > below the fixed limit D (Cover and Thomas 2006). Based on Feynman’s (2000) interpretation of information as a form of (free) energy, we can now construct a Boltzmann-like probability density in the “temperature” as
![$$\displaystyle{ dP(R,\mathcal{T} ) = \frac{\exp [-R/\mathcal{T} ]dR} {\int _{0}^{\infty }\exp [-R/\mathcal{T} ]dR} }$$](computational-psychiatry.files/image244.png)
(5.29)
since higher necessarily implies greater channel capacity.
The integral in the denominator is essentially a statistical mechanical partition function, and we define a “free energy” Morse Function F (Pettini 2007) as
![$$\displaystyle{ \exp [-F/\mathcal{T} ] =\int _{ 0}^{\infty }\exp [-R/\mathcal{T} ]dR = \mathcal{T} }$$](computational-psychiatry.files/image245.png)
(5.30)
so that ![$$F(\mathcal{T} ) = -\mathcal{T}\log [\mathcal{T} ]$$](computational-psychiatry.files/image246.png){kind=small}
.
Then an entropy-analog can also be defined as the Legendre transform of F:
(5.31)
As the usual first approximation, Onsager’s treatment of nonequilibrium thermodynamics (de Groot and Mazur 1984) can be applied, so that system dynamics are driven by the gradient of 
in essential parameters—here —under conditions of noise. This gives a stochastic differential equation
(5.32)
where μ is a “diffusion coefficient” representing the efforts of the underlying control mechanism, and β is the magnitude of an inherent impinging white noise dWt in the context of volatility, i.e., noise proportional to signal.
Again, via Jensen's inequality for a concave function, applying the Ito chain rule to 
in Eq. (5.32), a nonequilibrium steady state (nss) expectation for can be calculated as
(5.33)
Again, μ is interpreted as indexing the attempt by the embedding control apparatus to impose stability—raise . Thus impinging noise can significantly increase the probability that falls below the critical limit of Fig. 5.4, initiating a control failure.
However, 
is an expectation, so that, in this model, there is always some nonzero probability that will fall below the critical value α0 in the multimodal expression for : sporadic control dysfunctions have not been eliminated. Raising μ and lowering β decreases their probability, but will not drive it to zero in this model, a matter of some importance in determining population-rates of sleep disorders.
5.9 Chronic Dysfunctions of the Sleep Cycle
The normal sleep cycle involves regular transitions between the quasi-stable nonequilibrium steady states of Fig. 5.3, designated as the peak efficiency modes N, R, W. While we have described failure of the control system regulating that cycle in terms of the expectation relation of equation (5.33), the mechanism involved episodic disturbances, sporadic control dysfunctions. Here, we attempt to model chronic failure of regulation, which appears to have a somewhat different aspect.
One possible approach applies the “cognitive paradigm” of Atlan and Cohen (1998), who recognized that the immune response is not merely an automatic reflex, but involves active choice of a particular response to insult from a larger repertoire of possible responses. Choice reduces uncertainty and implies the existence of an underlying information source (Wallace 2012).
Given an information source associated with an inherently unstable, rapidly acting cognitive sleep regulation system—represented here as “dual” to it—a “natural” equivalence class algebra emerges by choosing different system origin states b0 and defining an equivalence relation for two subsequent states b m , bn at times m, n > 0 by the existence of high probability—and hence “meaningful”—paths connecting them to the same origin point. Disjoint partition by equivalence class, similar to orbit equivalence classes in dynamical systems, defines a groupoid that is to be associated with the cognitive process. Groupoids generalize of the group concept in that there is not necessarily a product defined for each possible element pair (Weinstein 1996).
The equivalence classes define a set of cognitive dual information sources available to the inherently unstable sleep regulation system, generating a groupoid. Each orbit corresponds to a transitive groupoid whose disjoint union is the full groupoid. Each subgroupoid is associated with a distinct dual information source, and larger groupoids must have richer dual information sources than smaller.
Take 
as the sleep cycle control system’s dual information source associated with the groupoid element Gi , and Y as the information source associated with an imposed environmental stress, again in a large sense. Earlier chapters detail how the regularities of environmental exposures imply the existence of an environmental information source.
It is again possible to construct a “free energy” Morse Function (Pettini 2007). Take 
as the joint uncertainty of the two information sources. A pseudoprobability can be written:
![$$\displaystyle{ P[H_{G_{i}}] = \frac{\exp [-H_{G_{i}}/\mathcal{T} ]} {\sum _{j}\exp [-H_{G_{j}}/\mathcal{T} ]} }$$](computational-psychiatry.files/image255.png)
(5.34)
Again, is the “temperature” from Eq. (5.27), via Γ(r1, …), and the sum is over all possible cognitive modes.
Another Morse Function, 
, can now be defined as
![$$\displaystyle{ \exp [-\mathcal{F}/\mathcal{T} ] \equiv \sum _{j}\exp [-H_{G_{j}}/\mathcal{T} ] }$$](computational-psychiatry.files/image257.png)
(5.35)
Since groupoids are extensions of more familiar symmetry groups, we argue by abduction that it is possible to apply an extension of Landau’s picture of phase transition (Pettini 2007). In Landau’s spontaneous symmetry breaking approach, phase transitions driven by temperature changes occur as alteration of system symmetry, with higher energies at higher temperatures being more symmetric. Transition between symmetries is punctuated in the temperature index under the Data Rate Theorem for unstable control systems. Usually, there are only a very limited number of possible phases.
Lowering of can lead to sudden, highly punctuated, decrease in the complexity of cognitive process possible within the sleep cycle control system, driving it into a kind of “ground state collapse” in which the sleep cycle is no long “normal,” within expected and accepted cultural patterns.
The driving force is the integrated environmental insult Γ. Increasing Γ is then equivalent to lowering the “temperature” , and the system passes from high symmetry “free flow” to different forms of “crystalline” structure—broken symmetries representing the punctuated onset of regulatory failure.
To reiterate, if κ2∕κ 4 ≫ κ1∕κ 3 < a0 in Eq. (5.27) and its extension by replacing ρ with the composite scalar Γ(r 1, …), accumulated environmental insult will quickly bring the effective “temperature” below some critical value, raising the probability for, or triggering the collapse into, a dysfunctional ground state.
Sufficient conditions for the intractability—stability—of the pathological state can be explored as follows.
Given a vector of parameters characteristic of and driving the pathological phase, J, measuring deviations from a nonequilibrium steady state, the “free energy” analog 
in Eq. (5.35) can be used to define a new “entropy” scalar as the Legendre transform
(5.36)
As before, a first order dynamic equation follows from a stochastic version of the Onsager formalism from nonequilibrium thermodynamics (de Groot and Mazur 1984)
(5.37)
where μi, k defines a diffusion matrix, the σi are parameters, and dBt represents a noise that may not the usual undifferentiated white noise.
Factoring out Ji , Eq. (5.37) takes the form
(5.38)
where Yt i is a stochastic process.
The expectation of J is found in terms of the Doleans-Dade exponential (Protter 1990):
![$$\displaystyle{ E(J_{t}^{i}) \propto \exp (Y _{ t}^{i} - 1/2[Y _{ t}^{i},Y _{ t}^{i}]) }$$](computational-psychiatry.files/image262.png){kind=small}
(5.39)
taking [Yt i , Y t i ] as the quadratic variation of the stochastic process Yt i (Protter 1990). Applying the Mean Value Theorem, if
![$$\displaystyle{ 1/2d[Y _{t}^{i},Y _{ t}^{i}]/dt> dY _{ t}^{i}/dt }$$](computational-psychiatry.files/image263.png){kind=small}
(5.40)
then the (culturally sculpted) pathological ground state is stable: deviations from nonequilibrium steady state measured by Jt i converge in expectation to 0. Thus sufficient noise—determining the quadratic variation—can lock-in the failure of the sleep cycle.
The argument is similar to that of ecosystem resilience theory (e.g., Holling 1973) characterizing multiple quasi-stable nonequilibrium steady states among interacting populations. Pristine alpine lakes, having limited nutrient inflows, can be permanently shifted into a toxic eutrophic state by excess nutrient influx—a sewage leak, fertilizer runoff, and so on. Once shifted, the lake’s ecology remains trapped in a dynamic mode of recurrent “red tide”-like plankton blooms even after sewage or fertilizer inflow ceases.
5.10 Discussion and Conclusions
Evolutionary process has selected for unstable control systems in higher animals that can react relatively swiftly to patterns of threat or affordance, but require strict ongoing regulation at different scales and levels of organization. Here, we have argued that consciousness, perhaps the most significant and sophisticated rapid large-scale neural process, must be supplied with high rates of metabolic free energy to both operate and stabilize the basic physiological dynamics. That is, both the “stream of consciousness” and the “riverbanks” that confine it to socially defined useful realms are constructed and reconstructed moment-by-moment in response to highly dynamic internal and environmental circumstances. High speed response requires considerable metabolic free energy.
Further, neural structures in higher animals are “coevolutionary” in that they respond rapidly both to incoming environmental signals, in a large sense, and to signals from other neural systems. It has long been known that stabilizing coevolutionary computing systems is as inherently difficult as programming them (Wallace 2017). Although working out the full details remains to be done, punctuated transitions seem inherent.
As has often been speculated, however, failure of regulation appears to underlie many psychiatric disorders.
Emotions, Thayer and Lane (2000) assert, are an integrative index of individual adjustment to changing environmental demands, a response to an environmental event that allows rapid mobilization of multiple subsystems. Emotions allow the efficient coordination of the organism for goal-directed behavior. When the system works properly, it allows for flexible adaptation to changing environmental demands. An emotional response must be regulated to represent a proper selection of an appropriate response and the inhibition of other less appropriate responses from a more or less broad behavioral repertoire of possible responses. From their perspective, disorders of affect represent a condition in which the individual is unable to select the appropriate response, or to inhibit the inappropriate response, so that the response selection mechanism is somehow corrupted—regulation fails.
Gilbert (2001) similarly suggests that a canonical form of such corruption is the inappropriate excitation of modes that, in other circumstances, represent normal evolutionary adaptations, again representing a fundamental failure of regulation.
The formal development thus extends the perspective of Wallace (2015) on the pathologies of mitochondrial dysfunction toward realms of psychiatric disorders.
Our argument further suggest a necessarily intimate synergism between sleep disorders and a spectrum of psychopathologies. That is, if the regulatory machinery for sleep is that of consciousness off-duty, as it were, sleep disorders imply some form of consciousness failure—psychopathology. Indeed, this has long been known. As Morin and Ware (1996) put it,
Epidemiological, cross sectional, and longitudinal data suggest a high rate of comorbidity between sleep disturbance and psychopathology , particularly between insomnia, anxiety, and depression. Between 50% and 80% of psychiatric patients complain of sleep disturbances during the acute phase of their illness. Conversely, among treatment-seeking individuals with a primary complaint of insomnia and randomly selected community samples, approximately one third display a concurrent psychopathology, one third exhibit psychological symptoms that do not necessarily exceed the threshold for a psychiatric disorder, and another third present insomnia as a functionally autonomous disorder. There is a positive relationship between severity of sleep disturbances and concurrent psychopathology…
More recent studies confirm the relation (e.g., Eidelman et al. 2012).
However, for humans, atomistic, individual-scale regulation must be iterated to include social and cultural influences. “Culture,” to use the words of the evolutionary anthropologist Robert Boyd, “is as much a part of human biology as the enamel on our teeth,” and this leads to extensions of the transition arguments above: the principal environment of humans is other humans, and we are the naked mole rats of primates. Thus social interaction and cultural heritage tend to confine individual consciousness to realms leading to socially acceptable phenotypes. Failure of such constraint is then socially constructed as misbehavior or pathology. Indeed, as described in Chap. 3, cultural norms and social interaction are generally synergistic with individual and group cognition and their disorders. Most particularly, a canonical failure mode in atomistic cultures is found to be a ground state collapse analogous to the psychopathic behaviors predicted by mainstream Western economic models. That is, high rates of psychopathic and antisocial personality disorders emerge as culture-bound syndromes characteristic of Westernizing societies, or of those undergoing large-scale social disintegration.
From the perspective of this chapter, in which consciousness-and-regulation are a synergistic composite, sleep must also reflect similar cultural influences, and this is indeed well known. That is, the structure, content, and meaning of sleep itself are very much cultural artifacts. Juillerat (1999), for example, describes sleep in the Yafar culture of Papua, New Guinea as follows:
The conceptions [of] the Yafar about the constituents of personhood and the transformations under which they go at death…[involve] a double, the spiritual value of blood and bones, and an Ego (or soul) that becomes autonomous during sleep and communicates with the spirits of the deceased during dreams…Death does not destroy them; it reorganizes them differently. Whereas the living person experiences a temporal alternation between being awake and asleep (conscious/unconscious), this cleavage stabilizes after death, when constituents are redistributed among different categories of spirits. The quality of relations that people maintain with spirits, as well as cosmological divisions, suggest a native metapsychology.
Jeong (1995), a Korean researcher, writes
Sleep…is a developmental product…subjected to the vicissitudes of human behavior and culture…sleep medicine/research [must involve]…a developmental perspective…Understanding of sleep and of sleep disorders is not complete without in-depth understanding of culture, philosophy, and tradition from developmental perspectives. Traditional ideas and wisdom from the past are the unavoidable resources for further understanding sleep and developing sleep research and medicine…
Worthman and Melby (2002), following a lengthy anthropological study of sleep ecologies across cultures, go further, concluding that
…[P]hysical, social, and temporal factors generating variation in human sleep ecology…may be paralleled by variation not only in sleep behavior but also in its physiology—specific cultural settings and practices may be associated with specific, distinctive risks for disorders of sleep and state regulation.
…[A]ssociations of cultural ecologies of sleep to such “basic” physiological regulatory systems as sleep biology, chronobiology, state regulation, and emotion regulation would imply that these systems are partially influenced or organized through cultural ecologies operating developmentally and across the life-span. Further, these associations may argue the need for attention to cultural ecology in the explanation, prevention, and possible treatment of disorders of these systems.
In sum, the composite of consciousness-and-regulation, its diurnal and other dynamics, and its pathologies, seem, for humans, as much cultural artifacts as they are biological realities. That is, an implication of our analysis is that the differences in cultural expression of waking consciousness that Nisbett et al. (2001) and other cultural psychologists have observed should be fully carried over into our understanding of all aspects of human psychology. Heine (2001) states the underlying case as follows:
Cultural psychology does not view culture as a superficial wrapping of the self, or as a framework within which selves interact, but as something that is intrinsic to the self. It assumes that without culture there is no self, only a biological entity deprived of its potential…Cultural psychology maintains that the process of becoming a self is contingent on individuals interacting with and seizing meanings from the cultural environment…
Other higher animals, for whom culture is a less central experience, may still display dynamics somewhat similar to what we have explored here, although inferences from experiments on them may not translate well into human modalities that have been strongly sculpted by embedding culture. For humans, social interactions carrying cultural structures must impose themselves both on long-term developmental trajectories and on the conformation and dynamics of the short-term regulators that define the riverbanks confining the streams of waking and REM consciousness.
We would argue that epigenetic information sources—including, but not limited to embedding/internalized culture—act as analogs to a tunable catalyst, directing development into different characteristic pathways according to the structure of external signals, a perspective having significant implications for understanding how environmental stressors, in a large sense, can induce a broad spectrum of developmental disorders in humans.
References
Aktipis, C., A. Boddy, G. Jansen, U. Hibner, M. Hochberg. C. Maley, and G. Wilkinson. 2015. Cancer across the tree of life: Cooperation and cheating in multicellularity. Philosophical Transactions B 370: 20140219.CrossRef
Atiyah, M., and I. Singer. 1963. The index of elliptical operators on compact manifolds. Bulletin of the American Mathematical Society 69: 322–433.CrossRef
Atlan, H., and I. Cohen. 1998. Immune information, self-organization and meaning. International Immunology 10: 711–717.CrossRefPubMed
Booth, V., and C. Diniz Behn. 2014. Physiologically-based modeling of sleep-wake regulatory networks. Mathematical Biosciences 250: 54–68.CrossRefPubMed
Borbely, A. 1982. A two process model of sleep regulation. Human Neurobiology 1: 195–204.PubMed
Champagnat, N., R. Ferriere, and S. Meleard. 2006. Unifying evolutionary dynamics: From individual stochastic process to macroscopic models. Theoretical Population Biology 69: 297–321.CrossRefPubMed
Christie, S.T., and P. Schrater. 2015. Cognitive cost as dynamic allocation of energetic resources. Frontiers in Neuroscience 9: 289 (online).CrossRefPubMedPubMedCentral
Clarke, D., and L. Sokoloff. 1999. Circulation and energy metabolism of the brain, Chapter 31. In Basic Neurochemistry: Molecular, Cellular and Medical Aspects, ed. G. Siegel et al., 6th ed. New York: Lippincott-Raven.
Cohen, I. 2000. Tending Adam’s Garden: Evolving the Cognitive Immune Self. New York: Academic Press.
Cover, T., and J. Thomas. 2006. Elements of Information Theory. New York: Wiley.
Daan, S., D. Beersma, and A. Borbely. 1984. Timing of human sleep: Recovery process gated by a circadian pacemaker. American Journal of Physics 246: 161–178.
de Groot, S., and P. Mazur. 1984. Non-equilibrium Thermodynamics. New York: Dover Publications.
Dembo, A., and O. Zeitouni. 1998. Large Deviations and Applications. New York: Springer.CrossRef
Eidelman, P., A. Gershon, E. McGlinchey, and A. Harvey. 2012. Sleep and psychopathology. In The Oxford Handbook of Sleep and Sleep Disorders, ed. G. Morin and C. Espie. New York: Oxford Handbooks.
Ellis, R. 1985. Entropy, Large Deviations, and Statistical Mechanics. New York: Springer.CrossRef
Evans, B. 2003. Sleep, consciousness and the spontaneous and evoked electrical activity in the brain. Is there a Cortical Integrating Mechanism? Neurophysiologie Clinique 33: 1–10.CrossRefPubMed
Feynman, R. 2000. Lectures on Computation. Boulder, CO: Westview Press.
Gilbert, P. 2001. Evolutionary approaches to psychopathology: The role of natural defenses. Australian and New Zealand Journal of Psychiatry 35: 17–27.CrossRefPubMed
Glazebrook, J.F., and R. Wallace. 2009. Rate distortion manifolds as model spaces for cognitive information. Informatica 33: 309–345.
Harris, J., R. Jolivet, and D. Attwell. 2012. Synaptic energy use and supply. Neuron 75: 762–777.CrossRefPubMed
Hazewinkel, M. 2002. Encyclopedia of Mathematics, ‘Index Formulas’. New York: Springer.
Heine, S. 2001. Self as cultural object: An examination of East Asian and North American elves. Journal of Personality 69: 881–906.CrossRefPubMed
Hobson, J.A. 2009. REM sleep and dreaming: Towards a theory of protoconsciousness. Nature Reviews Neuroscience 10: 803–813.PubMed
Holling, C. 1973. Resilience and stability of ecological systems. Annual Reviews of Ecological Systematics 4: 1–23.CrossRef
Hudson, A., D. Calderon, D. Pfaff, and A. Proekt. 2014. Recovery of consciousness is mediated by a network of discrete metastable activity states. PNAS 111: 9283–9288.CrossRefPubMedPubMedCentral
Jeong, D.U. 1995. Sleep in Korea: A developmental perspective. Psychiatry and Clinical Neurosciences 49: 115–117.CrossRefPubMed
Juillerat, B. 1999. Yafar death: Metamorphosis of the self in a Melanesian culture. Homme 149: 63–72 (in French).CrossRef
Khasminskii, R. 2012. Stochastic Stability of Differential Equations, 2nd ed. New York: Springer.CrossRef
Kostal, L., and R. Kobayashi. 2015. Optimal decoding and information transmission in Hodgkin-Huxley neurons under metabolic cost constraints. BioSystems. doi: 10.1016/j.biosystems.2015.06.008.PubMed
Laidler, K. 1987. Chemical Kinetics, 3rd ed. New York: Harper and Row.
Latimer, K., J. Yates, M. Meister, A. Huk, and J. Pillow. 2015. Single-trial spike trains in parietal cortex reveal discrete steps during decision-making. Science 349: 184–187.CrossRefPubMedPubMedCentral
Lee, J. 2000. Introduction to Topological Manifolds. New York: Springer.
Lu, J., D. Sherman, and C. Saper. 2006. A putative flip-flop switch for control of REM sleep. Nature 441: 589–594.CrossRefPubMed
Mackay, I., and N. Rose, eds. 2014. The Autoimmune Diseases, 5th ed. New York: Academic Press.
Madsen, P., and S. Vorstrup. 1991. Cerebral blood flow and metabolism during sleep. Cerebrovascular Brain Metabolism Review 3: 281–296.
Markman, A., and A.R. Otto. 2011. Cognitive systems optimize energy rather than information. Behavioral and Brain Sciences 34: 207.CrossRef
Mindwood, K., et al. 2004. Tissue repair and the dynamics of the extracellular matrix. International Journal of Biochemistry and Cell Biology 36: 1031–1037.CrossRef
Morin, C., and J. Ware. 1996. Sleep and psychopathology. Applied and Preventive Psychology 5: 211–224.CrossRef
Nair, G., F. Fagnani, S. Zampieri, and R. Evans. 2007. Feedback control under data rate constraints: An overview. Proceedings of the IEEE 95: 108–137.CrossRef
Nisbett, R., K. Peng, C. Incheol, and A. Norenzayan. 2001. Culture and systems of thought: Holistic vs. analytic cognition. Psychological Reviews 108: 291–310.CrossRef
Nunney, L. 1999. Lineage selection and the evolution of multistage carcinogenesis. Proceedings of the Royal Society B 266: 493–498.CrossRefPubMedPubMedCentral
Pettini, M. 2007. Geometry and Topology in Hamiltonian Dynamics. New York: Springer.CrossRef
Phillips, A., and P. Robinson. 2007. A quantitative model of sleep-wake dynamics based on the physiology of the brainstem ascending arousal system. Journal of Biological Rhythms 22: 167–179.CrossRefPubMed
Protter, P. 1990. Stochastic Integration and Differential Equations. New York: Springer.CrossRef
Rakoff-Nahoum, S. 2006. Why cancer inflammation? Yale Journal of Biology and Medicine 76: 123–130.
Rau, H., and T. Elbert. 2001. Psychophysiology of arterial baroreceptors and the etiology of hypertension. Biological Psychology 57: 179–201.CrossRefPubMed
Richerson, P., and R. Boyd. 2006. Not by Genes Alone: How Culture Transformed Human Evolution. Chicago: University of Chicago Press.
Ristroph, L., G. Ristroph, S. Morozova, A. Bergou, S. Chang, J. Guckenheimer, Z. Wang, and I. Cohen. 2013. Active and passive stabilization of body pitch in insect flight. Journal of the Royal Society Interface 10: 20130237.CrossRefPubMedCentral
Saper, C., G. Cano, and T. Scammell. 2005. Homeostatic, circadian, and emotional regulation of sleep. Journal of Comparative Neurology 493: 92–98.CrossRefPubMed
Schiff, N. 2008. Central thalamic contributions to arousal regulation and neurological disorders of consciousness. Annals of the New York Academy of Sciences 1129: 105–118.CrossRefPubMed
Sergeant C., and S. Dehaene. 2004. Is consciousness a gradual phenomenon? Psychological Science 15: 11–19.
Skeldon, A., D. Dijk, and G. Derks. 2014. Mathematical models for sleep-wake dynamics: Comparison of the two-process and a mutual inhibition neuronal model. Plos One 9/8/: e103877.
Thayer, J., and R. Lane. 2000. A model of neurovisceral integration in emotional regulation and dysregulation. Journal of Affective Disorders 61: 201–216.CrossRefPubMed
Wallace, R. 2012. Consciousness, crosstalk and the mereological fallacy: An evolutionary perspective. Physics of Life Reviews 9: 426–453.CrossRefPubMed
Wallace, R. 2015. An Information Approach to Mitochondrial Dysfunction: Extending Swerdlow’s Hypothesis. World Scientific Series in Information Studies, vol. 4. Singapore: World Scientific.
Wallace, R. 2017. An Information Treatment of Instabilities in Critical Systems. Singapore: World Scientific.
Wallace, R., and D. Wallace. 2009. Code, context, and epigenetic catalysis in gene expression. Transactions on Computational Systems Biology XI, LNABI, vol. 5750, 283–334. Lecture Notes in Bioinformatics, Lecture Notes in Computer Science. New York: Springer.
Wallace, R., and D. Wallace. 2010. Gene Expression and Its Discontents: The Social Production of Chronic Disease. New York: Springer.CrossRef
Weinstein, A. 1996. Groupoids: unifying internal and external symmetry. Notices of the American Mathematical Association 43: 744–752.
Worthman, C., and M. Melby. 2002. Toward a comparative developmental ecology of human sleep. In Adolescent Sleep Patterns: Biological, Social, and Psychological Influences, ed. M. Carskadon, 69–117. New York: Cambridge University Press.CrossRef